1,999 research outputs found
Some remarks on domination in cubic graphs
We study three recently introduced numerical invariants of graphs, namely, the signed domination number γs, the minus domination number γ- and the majority domination number γmaj. An upper bound for γs and lower bounds for γ- and γmaj are found, in terms of the order of the graph
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Domination number of graphs with minimum degree five
We prove that for every graph on vertices and with minimum degree
five, the domination number cannot exceed . The proof combines
an algorithmic approach and the discharging method. Using the same technique,
we provide a shorter proof for the known upper bound on the domination
number of graphs of minimum degree four.Comment: 17 page
New bounds on the signed total domination number of graphs
In this paper, we study the signed total domination number in graphs and
present new sharp lower and upper bounds for this parameter. For example by
making use of the classic theorem of Turan, we present a sharp lower bound on
this parameter for graphs with no complete graph of order r+1 as a subgraph.
Also, we prove that n-2(s-s') is an upper bound on the signed total domination
number of any tree of order n with s support vertices and s' support vertives
of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur
Maker-Breaker total domination game
Maker-Breaker total domination game in graphs is introduced as a natural
counterpart to the Maker-Breaker domination game recently studied by Duch\^ene,
Gledel, Parreau, and Renault. Both games are instances of the combinatorial
Maker-Breaker games. The Maker-Breaker total domination game is played on a
graph by two players who alternately take turns choosing vertices of .
The first player, Dominator, selects a vertex in order to totally dominate
while the other player, Staller, forbids a vertex to Dominator in order to
prevent him to reach his goal.
It is shown that there are infinitely many connected cubic graphs in which
Staller wins and that no minimum degree condition is sufficient to guarantee
that Dominator wins when Staller starts the game. An amalgamation lemma is
established and used to determine the outcome of the game played on grids.
Cacti are also classified with respect to the outcome of the game. A connection
between the game and hypergraphs is established. It is proved that the game is
PSPACE-complete on split and bipartite graphs. Several problems and questions
are also posed.Comment: 21 pages, 5 figure
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